Equalizer for Single Carrier FDMA Receiver

ABSTRACT

A method of equalizing a received signal compensates for frequency selectivity of the communication channel taking into account channel estimation errors. The method comprises generating channel estimates for the received signal, computing filter weights for an equalizer based on said channel estimates and a covariance of the channel estimation error, and filtering the received signal using the computed filter weights.

BACKGROUND

The present invention relates generally to equalization of receivedsignals in a mobile communication system and, more particularly, to anequalizer for a receiver in a single carrier frequency division multipleaccess (SC-FDMA) system.

Orthogonal Frequency-Division Multiple Access (OFDMA) is an attractivetechnique for sharing an available radio resource with multiple users ina high-speed wireless data communication system. Since each subcarrierof an OFDMA signal is simply scaled by a complex-valued scalar afterpassing through a time dispersive channel, demodulation can be performedfor each subcarrier individually, and hence equalization is not neededin the receiver. Moreover, through the use of a cyclic prefix,orthogonality among different subcarriers is preserved even if they arenot completely synchronized so long as the relative time delay islimited. This property is particularly desirable for uplinkcommunications because users assigned to different subcarriers aretypically only coarsely time aligned.

A major drawback of OFDMA is the high peak-to-average-power ratio (PAR),or equivalently, the high crest factor (CF) (square root of PAR) of thetransmitted waveform, which can cause undesired out-of-band radiationand/or inefficient power amplification in mobile terminals. Because ofthis limitation, Single-Carrier Frequency Division Multiple Access(SC-FDMA), whose transmitted waveforms have considerably lowerpeak-to-average-power ratio (PAR) than those of OFDMA, has recently beenselected by 3GPP as the standard access method for the uplink (UL) inEvolved UTRA. The low PAR property of SC-FDMA signals enables the mobileterminals to transmit at higher efficiency while reducing undesiredout-of-band emissions.

In an SC-FDMA system, there are two different methods of allocatingsubcarriers to different users, referred to as the localized or thedistributed allocations. The former method allocates contiguoussubcarriers to each individual user. This method requires less pilotoverhead for channel estimation but provides limited frequency diversityfor each user. The second method allocates subcarriers that are evenlydistributed over the spectrum assigned to each user. It provides morefrequency diversity but generally requires more pilot overhead forchannel estimation. Both carrier allocation methods result intransmitted signals that have significantly lower PAR than conventionalOFDMA signals.

Unlike conventional OFDMA systems, where the modulated symbolstransmitted over different frequency tones can be demodulatedindependently of other symbols at the receiver, SC-FDMA requires anequalizer at the receiver to compensate for the frequency selectivity ofthe channel in order to demodulate the transmitted symbols. Although itis well known that a time-domain maximum likelihood sequence estimation(MLSE) equalizer is optimal in this situation, the complexity of such anequalizer will be exorbitant for the high transmission rates expected inE-UTRA. Consequently, a reduced-complexity, suboptimal frequency-domainequalizer is needed for SC-FDMA.

SUMMARY

A method of equalizing a received signal compensates for frequencyselectivity of the communication channel while taking into accountchannel estimation errors. The method comprises generating channelestimates for the received signal, computing filter weights for anequalizer based on said channel estimates and a covariance of thechannel estimation error, and filtering the received signal using thecomputed filter weights.

In one exemplary embodiment, the equalizer may comprise a single stageequalization filter that uses error-compensated filter weights to filterthe received signal. In another embodiment, the equalizer may comprise aprefilter stage and an equalization stage. The received signal is firstfiltered in the prefilter to compensate for channel estimation errorsand subsequently filtered in the equalization stage to compensate forthe frequency selectivity of the channel.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a receiver for a single carrier FDMA system accordingto one exemplary embodiment of the present invention.

FIG. 2 illustrates a first embodiment of an equalizer for the receivershown in FIG. 1.

FIG. 3 illustrates a second embodiment of an equalizer for the receivershown in FIG. 1.

FIG. 4 illustrates an exemplary prefilter for the equalizer embodimentshown in FIG. 3.

DETAILED DESCRIPTION

Referring now to the drawings, FIG. 1 illustrates the main functionalelements of an SC-FDMA receiver 10 according to one exemplaryembodiment. The sampled signals from two or more antennas (not shown)are first converted into the frequency domain by a time-to-frequencyconverter 12. The time-to-frequency converter 12 converts the samplesinto the frequency domain using an N-point Fast Fourier Transform (FFT).After conversion into the frequency domain, a subcarrier mappingfunction 14 extracts the received signal on the subcarriers of interestfrom the FFT output, where N_(C) is the number of subcarriers. Thereceived signal R_(i) for the i^(th) antenna may be represented by thesignal vector R_(i)=(R_(i)[1], R_(i)[2], . . . , R_(i)[N_(c)])^(T), fori=1, 2, . . . , r. A typical baseband model of the received signalvector R_(i) for the i^(th) antenna is given by:

R _(i) =D(H _(i))S+V _(i),   (1)

where H_(i)=(H_(i)[1], H_(i)[2], . . . , H_(i)[N_(c)])^(T) denotes avector of channel coefficients over the desired frequency sub-carriersfor the i^(th) antenna, D(H_(i)) denotes a diagonal matrix with elementsof H_(i) as the diagonal elements, S=[S[1], S[2], . . . , S[N_(c)])^(T)represents the N_(C)-point FFT of the time-domain, modulated symbolsZ=[Z[1], Z[2], . . . , Z[N_(c)])^(T) such that ESS^(H)=I, andV_(i)=(V_(i)[1], V_(i)[2], . . . , V_(i)[N_(c)])^(T) denotes the noisevector at the i^(th) antenna, which is assumed to be a zero-mean,Gaussian-distributed random vector. The noise vector components areassumed to be uncorrelated across different subcarriers, i.e.,EV_(i)[k]V_(j)*[m]=0 if k≠m for any i and j.

A channel estimator 16 processes the received signals R_(i) to generateestimates Ĥ_(i) of the channel coefficients and provides the channelestimates Ĥ_(i) to a signal combiner 18. The signal combiner 18 combinesthe received signals R_(i) from each antenna i using the channelestimates to produce a combined signal vector R=(R[1], R[2], . . . ,R[N_(c)])^(T). The signal combiner 18 may comprise, for example, amaximal ratio combiner (MRC) or Interference Rejection Combiner (IRC).In the case when the noise is uncorrelated across antennas andidentically distributed across subcarriers, i.e., E[V_(i)V_(j)^(H)]=σ_(i) ²Iδ(i−j), an MRC can be used. For an MRC, the signalcombiner 18 combines the signal vectors R_(i) from each antennaaccording to:

$\begin{matrix}{{R = {\sum\limits_{i = 1}^{r}\frac{{D\left( G_{i} \right)}R_{i}}{\sigma_{i}}}},} & (2)\end{matrix}$

where G_(i)≡(G_(i)[1], G_(i)[2], . . . , G_(i)[N_(c)])^(T) represent thecombining weight for the i^(th) antenna, and D(G_(i)) comprises adiagonal matrix with the elements of G_(i) on its diagonal. Thecombining weights G_(i) may be calculated according to:

$\begin{matrix}{{{G_{i}\lbrack k\rbrack} = \frac{{\hat{H}}_{i}^{*}\lbrack k\rbrack}{\hat{H}\lbrack k\rbrack}},{{{where}\mspace{14mu} {\hat{H}\lbrack k\rbrack}} \equiv {\sqrt{\sum\limits_{i = 1}^{r}{{{\hat{H}}_{i}\lbrack k\rbrack}}^{2}}.}}} & (3)\end{matrix}$

In the case when the noise vectors at different antennas are correlated(i.e., the noise is spatially colored), the MRC may be replaced by anIRC in order to achieve the optimal performance, which is well-known tothose skilled in the art. For example, with the IRC the signal combiner18 combines the signal vectors R_(i) from each antenna according to:

R=(I _(N) _(c) {circle around (x)}1 _(r) ^(T))D( G )Λ _(V) ⁻¹ R .   (4)

In Equation (4):

R =vec([R ₁ , R ₂ , . . . R _(r)]^(T));

Λ _(V) ≡E└ VV ^(H)┘ where V =vec([V ₁ , V ₂ , . . . , V _(r)]^(T));

G =([G ₁[1], G ₂[1], . . . , G _(r)[1], G ₁[2], G ₂[2], . . . , G_(r)[2], . . . , G ₁ [N _(c) ], G ₂ [N _(c) ], . . . , G _(r) [N_(c)]]^(T))

where

${G_{i}\lbrack k\rbrack} = \frac{{\hat{H}}_{i}^{*}\lbrack k\rbrack}{\hat{H}\lbrack k\rbrack}$

and Ĥ[k] is the square root of the k^(th) diagonal element of the matrix(I_(N) _(c) {circle around (x)}1 _(r) ^(T))D( H)^(H) Λ _(V) ⁻¹D(H)(I_(N) _(c) {circle around (x)}1 _(r)) and

H=vec([H ₁ , H ₂ , . . . , H _(r))^(T));

1_(r)≡(1, 1, . . . , 1)^(T) denotes an all-one column vector of lengthr;

I_(N) _(c) denotes an N_(c)×N_(c) identity matrix;

vec(·) denotes the vectorization operation of stacking the columns ofthe argument; and

{circle around (x)} denotes the Kronecker product.

The combined signal vector R can then be modeled as:

R=D(Ĥ)S+V,   (5)

where Ĥ=(Ĥ[1], Ĥ[2], . . . , Ĥ[N_(c)])^(T) denotes a vector ofequivalent channel coefficients over the desired frequency sub-carriersafter combining, D(Ĥ) denotes a diagonal matrix with elements Ĥ of asthe diagonal elements, and V=(V[1], V[2], . . . , V[N_(c)])^(T) denotesa zero-mean Gaussian noise vector with covariance matrix E[VV^(H)]=I.

The combined signal vector R is input to a frequency domain equalizer20, which compensates the received signal vector R for the frequencyselectivity of the uplink channel. A weight calculator 26 receives thechannel estimates Ĥ_(i) for each antenna from the channel estimator 16and computes filter weights {circumflex over (F)} for the equalizer 20.The filter weight calculation is performed in a manner that takes intoaccount channel estimation errors. While it is not possible to computedirectly the channel estimation error, the covariance of the channelestimation error can be computed and used to refine the filter weightcalculation. A frequency-to-time converter 28 converts the equalizedsignal back into the time domain. The output of the frequency-to-timeconverter 28 is an estimate {circumflex over (Z)} of the QAM modulatedsymbols Z. A demodulator 30 and decoder 32 follow the frequency to timeconverter 28 for demodulating and decoding {circumflex over (Z)} toobtain an estimate of an original information signal I that wastransmitted

In a conventional receiver, filter weights {circumflex over (F)}_(CONV)for the equalizer can be computed according to:

{circumflex over (F)} _(conv) =D(Ĥ)^(H)(D(Ĥ)D(Ĥ)^(H) +I)⁻¹ =D(f),   (6)

where f=(f[1], f[2], . . . , f[N_(c)])^(T) and

$\begin{matrix}{{f\lbrack k\rbrack} = {\frac{{\hat{H}\lbrack k\rbrack}^{*}}{{{\hat{H}\lbrack k\rbrack}}^{2} + 1}.}} & (7)\end{matrix}$

The conventional way of computing the equalizer weights, as described inEquation (6), does not take into the account the channel estimationerror. The resulting filter weights {circumflex over (F)}_(CONV)therefore do not minimize the mean squared error between the transmittedsymbols and equalizer output when there are channel estimation errors.

According to one embodiment of the present invention, channel estimationerrors are taken into account to compute the filter weights {circumflexover (F)} for an error-compensated MMSE equalizer. If e=[e[1], e[2], . .. , e[N_(c)])^(T) denotes the channel estimation error such that H=Ĥ+e.The covariance of the channel estimation error e can then be given by:

Λ_(e)=Eee^(H).   (8)

The filter weight {circumflex over (F)} can then be modeled by:

$\begin{matrix}\begin{matrix}{\hat{F} = {\underset{F}{\arg \; \min}{{S - {FR}}}^{2}}} \\{= {\left( {ESR}^{H} \right)\left( {ERR}^{H} \right)^{- 1}}} \\{= {\left( {{ES}\left\lbrack {{{D\left( {\hat{H} + e} \right)}S} + V} \right\rbrack} \right)\left( {E\left\lbrack {{{D\left( {\hat{H} + e} \right)}S} + V} \right\rbrack} \right.}} \\\left. \left\lbrack {{D\left( {\hat{H} + e} \right)S} + V} \right\rbrack^{H} \right)^{- 1} \\{= {{D\left( \hat{H} \right)}^{H}\left( {{{D\left( \hat{H} \right)}{D\left( \hat{H} \right)}^{H}} + {E\left\lbrack {{eSS}^{H}e^{H}} \right\rbrack} + I} \right)^{- 1}}} \\{= {D\left( \hat{H} \right)^{H}\left( {{{D\left( \hat{H} \right)}{D\left( \hat{H} \right)}^{H}} + {E_{e}\left\{ {{{eE}\left\lbrack {{SS}^{H}\left. e \right\rbrack e^{H}} \right\rbrack} + I} \right)^{- 1}}} \right.}} \\{= {D\left( \hat{H} \right)^{H}\left( {{{D\left( \hat{H} \right)}{D\left( \hat{H} \right)}^{H}} + {E_{e}\left\{ {{{eE}\left\lbrack {SS}^{H} \right\rbrack}e^{H}} \right\rbrack} + I} \right)^{- 1}}} \\{{= {D\left( \hat{H} \right)^{H}\left( {{{D\left( \hat{H} \right)}{D\left( \hat{H} \right)}^{H}} + I + \Lambda_{e}} \right)^{- 1}}},}\end{matrix} & (9)\end{matrix}$

where it is assumed that the transmitted symbols S are independent ofthe channel estimation error e. The estimation error covariance matrixΛ_(e) can often be pre-computed according to the channel estimationmethod. For example, for maximum-likelihood channel estimator, it can beshown that:

$\begin{matrix}\begin{matrix}{\Lambda_{e} = {{E\left( {\hat{H} - H} \right)}\left( {\hat{H} - H} \right)^{H}}} \\{= {{W_{N}\left( {J,I} \right)}\left( {{W_{N}\left( {J,I} \right)}^{H}{D(P)}^{H}{D(P)}{W_{N}\left( {J,I} \right)}} \right)^{- 1}}} \\{{{W_{N}\left( {J,I} \right)}^{H},}}\end{matrix} & (10)\end{matrix}$

where P=[P[1], P[2], . . . , P[N_(c)])^(T) denotes the vector of pilotsymbols over which the channel is estimated, J denotes the index setcontaining indices of the N_(c) desired sub-carriers, I denotes theindex set containing the indices of the estimated channel tap locations,and W_(N) (J,I) denotes a sub-matrix of the N-point FFT matrix W_(N)formed by the rows indexed by the set J and the columns indexed by theset I . When the pilot symbols have a constant magnitude in frequencydomain, as it is the case when the pilot symbols are properly designed,Equation (10) reduces to:

Λ_(e) =W _(N)(J,I)(W _(N)(J,I)^(H) W _(N)(J,I)³¹ ¹ W _(N)(J,I)^(H).  (11)

Furthermore, in the case of distributed sub-carrier allocation where theindices in the set J are uniformly distributed over the N possibleindices, Equation (11) reduces further to

Λ_(e) =W _(N)(J,I)W _(N)(J,I)^(H).   (12)

The filter weights {circumflex over (F)} can be computed according toEquation (9) using one of Equations (10), (11), and (12) to compute thecovariance Λ_(e) of the channel estimation error. In this case, theequalizer 20 may comprise a single MMSE equalization filter 22 where thefilter coefficients are given by the filter weights {circumflex over(F)} as shown in FIG. 2.

It may be noted that Equation (9) can be rewritten as follows:

{circumflex over (F)}=D(Ĥ)^(H)(D(Ĥ)D(Ĥ)^(H) +I)⁻¹(D(Ĥ)D(Ĥ)^(H)+I)(D(Ĥ)D(Ĥ)^(H) +I+Λ _(e))⁻¹.   (13)

Note that the first term D(Ĥ)^(H)(D(Ĥ)D(Ĥ)^(H)+I)⁻¹ in Equation (13) isthe same as Equation (6). Therefore, Equation (13) can be reduced to:

{circumflex over (F)}={circumflex over (F)}_(conv){circumflex over (P)},  (14)

where

{circumflex over (P)}≡(D(Ĥ) D(Ĥ) ^(H) +I)(D(Ĥ)D(Ĥ)^(H) +I+Λ _(e))⁻¹  (15)

Equation (15) can also be rewritten as follows:

$\begin{matrix}\begin{matrix}{\hat{P} = {\left( {{{D\left( \hat{H} \right)}{D\left( \hat{H} \right)}^{H}} + I + \Lambda_{e} - \Lambda_{e}} \right)\left( {{{D\left( \hat{H} \right)}{D\left( \hat{H} \right)}^{H}} + I + \Lambda_{e}} \right)^{- 1}}} \\{= {I - {\Lambda_{e}\left( {{{D\left( \hat{H} \right)}{D\left( \hat{H} \right)}^{H}} + I + \Lambda_{e}} \right)}^{- 1}}} \\{{= {I - B}},}\end{matrix} & (16)\end{matrix}$

where B≡Λ_(e) (D(Ĥ)D(Ĥ)^(H)+I+Λ)⁻¹. It follows from Equation (16) thatan error-compensated MMSE equalizer can be implemented as a pre-filter24 followed by the conventional MMSE equalizer filter 26, as shown inFIG. 3. The filter weights {circumflex over (P)} for the pre-filter 24are a function of the estimated channel Ĥ and the error covariancematrix Λ_(e). The filter weights {circumflex over (F)}_(conv) for theconventional MMSE equalizer are a function of the channel estimates Ĥ.

FIG. 4 illustrates one exemplary embodiment of the pre-filter 24. Inthis embodiment, the received signal is passed through a compensationfilter 34 whose coefficients are given by B to obtain an estimate of thesignal contribution due to the channel estimate error. Subtractor 36subtracts the output of the compensation filter 28 from the receivedsignal R before passing through the conventional MMSE equalizer 22.

Those skilled in the art will appreciate that the inventive receiver 10can be implemented with a digital signal processor by executing codestored in a memory. The received signals on each antenna may bedownconverted to baseband, sampled, and digitized for input to thereceiver 10.

1. A method of equalizing a received signal, said method comprising:generating channel estimates for the received signal; computing filterweights for an equalizer based on said channel estimates and acovariance of the channel estimation error; and filtering the receivedsignal using the computed filter weights.
 2. The method of claim 1wherein computing filter weights comprises computing equalization filterweights based on said channel estimates and said covariance of saidchannel estimation error.
 3. The method of claim 2 wherein filtering thereceived signal comprises filtering the received signal in a singlefilter stage using said equalization filter weights.
 4. The method ofclaim 2 wherein computing filter weights for said equalizer comprisescomputing equalizer filter weights based on said channel estimates, andcomputing prefilter weights based on said channel estimates and saidcovariance of said channel estimation error.
 5. The method of claim 2wherein filtering the received signal comprises filtering the receivedsignal in successive filter stages using said prefilter weights and saidequalization filter weights.
 6. The method of claim 5 comprisingfiltering the received signal in a prefilter stage to estimate thecontribution attributable to the channel estimation error to thereceived signal, and subtracting the contribution of said channelestimation error to obtain an error compensated signal.
 7. The method ofclaim 6 comprising filtering the error compensated signal in anequalization stage to compensate for frequency selectivity of thechannel.
 8. The method of claim 1 further comprising combining signalsreceived on two or more antennas to obtain said received signal.
 9. Themethod of claim 8 wherein combining signals received on two or moreantennas comprises combining said received signals using maximal ratiocombining.
 10. The method of claim 8 wherein combining signals receivedon two or more antennas comprises combining said received signals usinginterference rejection combining.
 11. A receiver including an equalizerto equalize a received signal, said receiver comprising: a channelestimator to generate channel estimates for the received signal; anequalizer to filter the received signal; and a weight calculator tocalculate filter weights for the equalizer based on said channelestimates and a covariance of the channel estimation error.
 12. Thereceiver of claim 11 wherein the equalizer includes an error-compensatedequalization filter, and wherein the weight calculator computes filterweights for the equalization filter based on said channel estimates andsaid covariance of said channel estimation error.
 13. The receiver ofclaim 11 wherein the equalization filter comprises an MMSE filter. 14.The receiver of claim 11 wherein the equalizer includes a prefilter tofilter the received signal to obtain an error compensated signal, and anequalization filter to filter the error compensated signal to compensatefor frequency selectivity of the channel.
 15. The receiver of claim 14wherein the weight calculator computes filter weights for saidpre-filter based on said channel estimates and said covariance of saidchannel estimation error and computes filter weights for theequalization filter based on said channel estimates.
 16. The receiver ofclaim 15 wherein the pre-filter includes a compensation filter to filterthe received signal to estimate of the contribution attributable to thechannel estimation error to the received signal, and a subtractor tosubtract the estimated contribution of the channel estimation error fromthe received signal to obtain the error compensated signal.
 17. Thereceiver of claim 16 wherein the equalization filter comprises an MMSEfilter to filter the error compensated signal.
 18. The receiver of claim11 further comprising a combiner to combine signals received on two ormore antennas to obtain said received signal.
 19. The method of claim 18wherein the combiner comprises a maximal ratio combiner.
 20. The methodof claim 18 wherein the combiner comprises an interference rejectioncombiner.